# Beginner question: Notation for polynomial rings

I hope people will indulge me a newbie question on notation.

For reference, the textbook in question is Contemporary Abstract Algebra by Joseph Gallian (8th edition).

I was reading the chapter on extension fields, which defines the Fundamental Theorem of Field Theory as follows: "Let F be a field and let f(x) be a nonconstant polynomial in F[x]. Then there is an extension field E of F in which f(x) has a zero." It then gives the following example:

Let $f(x)=x^2 + 1 \in Q[x]$. Then, viewing $f(x)$ as an element of $E[x] = (Q[x]/\langle x^2 + 1\rangle)[x]$, we have $$f(x + \langle x^2 + 1 \rangle) = (x + \langle x^2 + 1 \rangle)^2 + 1 = x^2 + \langle x^2 + 1 \rangle + 1 = x^2 + 1 + \langle x^2 + 1 \rangle = 0 + \langle x^2 + 1 \rangle$$

I have to admit I'm slightly baffled by this example and his notation. So I understand that $Q[x]$ is a polynomial ring with rational coefficients, that $G/H$ is a factor/quotient ring (or a factor group in group theory), and that $\langle a \rangle$ is an ideal generated by $a$ (or a cyclic group generated by $a$ in group theory), but I'm having trouble understanding what this actually means when put together. Can someone help me understand this example? I'm having particular difficulty understanding what $E[x] = (Q[x]/\langle x^2 + 1\rangle)[x]$ means. What does this actually consist of (i.e. what are its elements)?

• They should at the very least have said $(\Bbb Q[t]/\langle t^2+1\rangle)[x]$ and say that $f(t+\langle t^2+1\rangle)=0$. Using $x$ for two different things is not a good thing to do. Dec 3, 2016 at 22:12
• @Arthur I agree, the way they wrote it is very confusing. Dumb question - $Q[t]/ \langle t^2 + 1\rangle$ is still a factor ring, right? Dec 3, 2016 at 22:45
• Is the variable name clash the only source of your confusion, i.e. if we instead write $\, E = \Bbb Q[t]/(t^2+1)\,$ then does that clear up the confusion? Dec 3, 2016 at 22:53

I must say the notations are rather ambiguous. I would use another indeterminate, say $t$ and set $E=\mathbf Q[t]/(t^2+1)$. It is a field extension of $\mathbf Q$, and in this field, the polynomial $f(x)=x^2+1$ has a root, and this root is $t$ (or $-t$), since $$f(t)=t^2+1\equiv 0\mod t^2+1.$$
• Is there a difference between $Q[t]/ \langle t^2 + 1\rangle$ and $Q[t]/(t^2 + 1)$? I've seen both notations used. Can both (a) and $\langle a \langle$ be used to denote an ideal generated by $a$? Dec 3, 2016 at 22:46
• @EJoshuaS: The notations usually have the same meaning (both $(a)$ and $\langle a\rangle$ mean the ideal generated by $a$), though you may occasionally encounter authors who give one a different meaning in some special context. In my experience $(a)$ is the more common notation. Dec 3, 2016 at 22:51
• @EJoshuaS The notation $\,(a,b)\,$ for ideals better emphasizes the important analogy with gcds (I don't ever recall seeing gcds written in angle bracket form). It's also consistent with $\!\pmod{a,b}$ notation (just omit the mod for more ideal notation!). The analogies are clarified when one studies divisor theory (modernizations of Kronecker's alternative to Dedekind's ideal theory). Dec 3, 2016 at 23:02